List of top Mathematics Questions on Probability Distributions

For a function \( f(x) = -x^2 - 2x + 30 \), which of the following statements are TRUE?
(A) \( f(x) \) is increasing on \( (-\infty, -1) \)
(B) \( f(x) \) is increasing on \( (-1, \infty) \)
(C) \( f(x) \) is decreasing on \( (-\infty, -1) \)
(D) \( f(x) \) is decreasing on \( (-1, \infty) \)
If a machine is correctly set up, it produces 80% acceptable items. If it is incorrectly set up, it produces only 30% acceptable items. From the past experience it was known that 90% of the setups are correctly done. If after a certain setup, the machine produces 2 acceptable items then the probability that the machine was correctly set up, is:

A random variable \( X \) has the following probability distribution table: 

\( X \)012
\( P(X) \)\( k \)\( 2k \)\( 3k \)

Find the exact value of the unknown parameter \( k \).

The probability distribution of a random variable X is given by

X012
P(X)\(1 - 7a^2\)\(\tfrac{1}{2}a + \tfrac{1}{4}\)\(a^2\)


If \(a > 0\), then \(P(0 < X \leq 2)\) is equal to

The probability distribution of a random variable X is given by
\begin{tabular}{|c|c|c|c|} \hline X & 0 & 1 & 2 \\ \hline P(X) & $1 - 7a^2$ & $\frac{1}{2}a + \frac{1}{4}$ & $a^2$ \\ \hline \end{tabular}
If a > 0, then P(0 $<$ x $\le$ 2) is equal to